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Properties

Label 11.3.243...976.4
Degree $11$
Signature $[3, 4]$
Discriminant $2.439\times 10^{23}$
Root discriminant \(133.69\)
Ramified primes $2,3,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PSL(2,11)$ (as 11T5)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^11 - 22*x^9 - 66*x^8 + 1276*x^6 + 5038*x^5 + 11264*x^4 + 14168*x^3 + 2156*x^2 - 7744*x + 224)
 
Copy content gp:K = bnfinit(y^11 - 22*y^9 - 66*y^8 + 1276*y^6 + 5038*y^5 + 11264*y^4 + 14168*y^3 + 2156*y^2 - 7744*y + 224, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 22*x^9 - 66*x^8 + 1276*x^6 + 5038*x^5 + 11264*x^4 + 14168*x^3 + 2156*x^2 - 7744*x + 224);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 22*x^9 - 66*x^8 + 1276*x^6 + 5038*x^5 + 11264*x^4 + 14168*x^3 + 2156*x^2 - 7744*x + 224)
 

\( x^{11} - 22x^{9} - 66x^{8} + 1276x^{6} + 5038x^{5} + 11264x^{4} + 14168x^{3} + 2156x^{2} - 7744x + 224 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $11$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[3, 4]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(243920281203464276606976\) \(\medspace = 2^{16}\cdot 3^{4}\cdot 11^{16}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(133.69\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{11/6}3^{1/2}11^{84/55}\approx 240.40284291526194$
Ramified primes:   \(2\), \(3\), \(11\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{42\cdots 64}a^{10}-\frac{90384852556237}{21\cdots 32}a^{9}+\frac{47983806308511}{21\cdots 32}a^{8}+\frac{393521179187369}{21\cdots 32}a^{7}+\frac{43182459194053}{10\cdots 66}a^{6}+\frac{478426516078439}{10\cdots 66}a^{5}-\frac{950665382024281}{21\cdots 32}a^{4}+\frac{61827835156621}{10\cdots 66}a^{3}-\frac{12485105883792}{529618963817633}a^{2}-\frac{45581671505467}{10\cdots 66}a+\frac{26007605918769}{529618963817633}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $6$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{10\cdots 29}{42\cdots 64}a^{10}+\frac{13\cdots 74}{529618963817633}a^{9}-\frac{27\cdots 51}{21\cdots 32}a^{8}-\frac{15\cdots 57}{21\cdots 32}a^{7}+\frac{31\cdots 55}{529618963817633}a^{6}+\frac{74\cdots 17}{10\cdots 66}a^{5}+\frac{66\cdots 11}{21\cdots 32}a^{4}+\frac{31\cdots 62}{529618963817633}a^{3}+\frac{89\cdots 23}{529618963817633}a^{2}-\frac{35\cdots 37}{10\cdots 66}a+\frac{51\cdots 75}{529618963817633}$, $\frac{127566538935193}{42\cdots 64}a^{10}-\frac{131237144846368}{529618963817633}a^{9}-\frac{174959089096337}{21\cdots 32}a^{8}+\frac{65\cdots 53}{21\cdots 32}a^{7}+\frac{17\cdots 93}{529618963817633}a^{6}+\frac{221931404629305}{10\cdots 66}a^{5}-\frac{14\cdots 17}{21\cdots 32}a^{4}-\frac{11\cdots 05}{529618963817633}a^{3}-\frac{46\cdots 15}{529618963817633}a^{2}+\frac{13\cdots 05}{10\cdots 66}a-\frac{18\cdots 61}{529618963817633}$, $\frac{45127455143177}{10\cdots 66}a^{10}-\frac{233657106780186}{529618963817633}a^{9}+\frac{391612728425114}{529618963817633}a^{8}+\frac{728367014753835}{10\cdots 66}a^{7}+\frac{78\cdots 27}{529618963817633}a^{6}-\frac{14\cdots 62}{529618963817633}a^{5}-\frac{67\cdots 33}{529618963817633}a^{4}-\frac{22\cdots 65}{529618963817633}a^{3}-\frac{49\cdots 54}{529618963817633}a^{2}-\frac{24\cdots 14}{529618963817633}a+\frac{30\cdots 89}{529618963817633}$, $\frac{32\cdots 21}{42\cdots 64}a^{10}-\frac{10\cdots 44}{529618963817633}a^{9}-\frac{84\cdots 51}{21\cdots 32}a^{8}+\frac{10\cdots 55}{21\cdots 32}a^{7}+\frac{34\cdots 57}{529618963817633}a^{6}-\frac{32\cdots 09}{10\cdots 66}a^{5}-\frac{42\cdots 05}{21\cdots 32}a^{4}-\frac{20\cdots 73}{529618963817633}a^{3}-\frac{51\cdots 36}{529618963817633}a^{2}+\frac{21\cdots 39}{10\cdots 66}a-\frac{31\cdots 51}{529618963817633}$, $\frac{212129655084324}{529618963817633}a^{10}-\frac{724316463296915}{10\cdots 66}a^{9}-\frac{81\cdots 63}{10\cdots 66}a^{8}-\frac{14\cdots 53}{10\cdots 66}a^{7}+\frac{12\cdots 97}{529618963817633}a^{6}+\frac{25\cdots 61}{529618963817633}a^{5}+\frac{63\cdots 07}{529618963817633}a^{4}+\frac{12\cdots 34}{529618963817633}a^{3}+\frac{75\cdots 60}{529618963817633}a^{2}-\frac{93\cdots 47}{529618963817633}a+\frac{25\cdots 19}{529618963817633}$, $\frac{15\cdots 87}{42\cdots 64}a^{10}+\frac{19\cdots 25}{10\cdots 66}a^{9}+\frac{15\cdots 59}{21\cdots 32}a^{8}-\frac{65\cdots 19}{21\cdots 32}a^{7}-\frac{97\cdots 08}{529618963817633}a^{6}-\frac{56\cdots 81}{10\cdots 66}a^{5}-\frac{21\cdots 59}{21\cdots 32}a^{4}-\frac{54\cdots 20}{529618963817633}a^{3}-\frac{28\cdots 22}{529618963817633}a^{2}+\frac{58\cdots 17}{10\cdots 66}a-\frac{85\cdots 35}{529618963817633}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 306439634.482 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 306439634.482 \cdot 1}{2\cdot\sqrt{243920281203464276606976}}\cr\approx \mathstrut & 3.86812450437 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^11 - 22*x^9 - 66*x^8 + 1276*x^6 + 5038*x^5 + 11264*x^4 + 14168*x^3 + 2156*x^2 - 7744*x + 224) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^11 - 22*x^9 - 66*x^8 + 1276*x^6 + 5038*x^5 + 11264*x^4 + 14168*x^3 + 2156*x^2 - 7744*x + 224, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 22*x^9 - 66*x^8 + 1276*x^6 + 5038*x^5 + 11264*x^4 + 14168*x^3 + 2156*x^2 - 7744*x + 224); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 22*x^9 - 66*x^8 + 1276*x^6 + 5038*x^5 + 11264*x^4 + 14168*x^3 + 2156*x^2 - 7744*x + 224); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$\PSL(2,11)$ (as 11T5):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 sibling: 12.0.140498081973195423325618176.2
Arithmetically equivalent sibling: 11.3.243920281203464276606976.5
Minimal sibling: 11.3.243920281203464276606976.5

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.6.11a1.1$x^{6} + 2$$6$$1$$11$$D_{6}$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.3.1.0a1.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
3.3.2.3a1.1$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(11\) Copy content Toggle raw display 11.1.11.16a2.1$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$$[\frac{8}{5}]_{5}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)